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Mirrors > Home > MPE Home > Th. List > 3sstr3g | Structured version Visualization version GIF version |
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.) |
Ref | Expression |
---|---|
3sstr3g.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
3sstr3g.2 | ⊢ 𝐴 = 𝐶 |
3sstr3g.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3sstr3g | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr3g.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 3sstr3g.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3sstr3g.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | sseq12i 3996 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | sylib 219 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ⊆ wss 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3942 df-ss 3951 |
This theorem is referenced by: complss 4122 uniintsn 4906 fpwwe2lem13 10053 hmeocls 22306 hmeontr 22307 usgrumgruspgr 26893 chsscon3i 29166 pjss1coi 29868 mdslmd2i 30035 satffunlem2lem2 32551 ssbnd 34949 bnd2lem 34952 trclubgNEW 39858 nzss 40529 |
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